Sex differences in weight gain (original Lord formulation)

Variables

weight time 1
weight time 2
weight change
sex

Issues

t-test on change vs. ancova
Statistician 1 concludes no difference in change
Statistician 2 concludes a difference in time 2 if controlling for time 1

Results

Both statisticians are correct
t-test on change = total effect
ancova = direct effect

DAG

Indirect effect: is a*c

Direct effect: b

Total effect: b + a*c - a

Data Example

library(dplyr)
group  = rep(c('B', 'A'), e=5)
pre = c(20,10,60,20,10,50,10,40,20,10)
post = c(70,50,90,60,50,20,10,30,50,10)

df = data.frame(id=factor(1:10), group, pre, post)
change = post-pre

dflong = tidyr::gather(df, key=time, value=score, pre:post) %>% arrange(id)
head(df)

  id group pre post
1  1     B  20   70
2  2     B  10   50
3  3     B  60   90
4  4     B  20   60
5  5     B  10   50
6  6     A  50   20

head(dflong)

  id group time score
1  1     B  pre    20
2  1     B post    70
3  2     B  pre    10
4  2     B post    50
5  3     B  pre    60
6  3     B post    90

It’s an extremely large group difference.

mod = "
  pre ~ a*group
  post ~ b*group + c*pre
  # change ~ -1*pre + 1*post
 
  # total effect
  TE := (b + a*c) - a
"

library(lavaan)
lpmod = sem(mod, data=df)

summary(lpmod)

lavaan (0.5-20) converged normally after  40 iterations

  Number of observations                            10

  Estimator                                         ML
  Minimum Function Test Statistic                0.000
  Degrees of freedom                                 0

Parameter Estimates:

  Information                                 Expected
  Standard Errors                             Standard

Regressions:
                   Estimate  Std.Err  Z-value  P(>|z|)
  pre ~                                               
    group      (a)   -2.000   11.027   -0.181    0.856
  post ~                                              
    group      (b)   41.053    7.490    5.481    0.000
    pre        (c)    0.526    0.214    2.454    0.014

Variances:
                   Estimate  Std.Err  Z-value  P(>|z|)
    pre             304.000  135.953    2.236    0.025
    post            139.789   62.516    2.236    0.025

Defined Parameters:
                   Estimate  Std.Err  Z-value  P(>|z|)
    TE               42.000    9.121    4.605    0.000

summary(lm(change~group, df))  # t-test on change scores = total effect


Call:
lm(formula = change ~ group, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
   -28     -6      0      2     32 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   -2.000      7.211  -0.277  0.78854   
groupB        42.000     10.198   4.118  0.00335 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 16.12 on 8 degrees of freedom
Multiple R-squared:  0.6795,    Adjusted R-squared:  0.6394 
F-statistic: 16.96 on 1 and 8 DF,  p-value: 0.003351

summary(lm(post~group+pre, df)) # 'ancova' uncovers direct effect etc.


Call:
lm(formula = post ~ group + pre, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-16.632  -6.368  -3.737   4.947  29.158 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  10.3158     9.1840   1.123  0.29838   
groupB       41.0526     8.9522   4.586  0.00253 **
pre           0.5263     0.2563   2.054  0.07912 . 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 14.13 on 7 degrees of freedom
Multiple R-squared:  0.776, Adjusted R-squared:  0.712 
F-statistic: 12.12 on 2 and 7 DF,  p-value: 0.005321

Aside

Use change score adjusting for pre. Identical result, but where coef for pre = ancova coef for pre -1


Call:
lm(formula = change ~ group + pre, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-16.632  -6.368  -3.737   4.947  29.158 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  10.3158     9.1840   1.123  0.29838   
groupB       41.0526     8.9522   4.586  0.00253 **
pre          -0.4737     0.2563  -1.848  0.10706   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 14.13 on 7 degrees of freedom
Multiple R-squared:  0.7846,    Adjusted R-squared:  0.7231 
F-statistic: 12.75 on 2 and 7 DF,  p-value: 0.004638

Birth Weight Paradox

Variables

birthweight
smoking mom
infant mortality

Issues

No difference score
Before, focus on clash between two seemingly legitimate methods of analysis
Now using ancova but results seem implausible

Results

low birthweight children have higher mortality rate (100 fold higher)

children of smoking mothers notably more likely to have low birghtweight

low birthweight children born to smoking mothers have a lower mortality rate

Conclusion: expectant mothers should start smoking!

Results

Collider bias (explain away effect)

Explanation

Perspective 1

Controlling just for smoking leaves other causes, resulting in bias
Controlling for smoking changes the probability of other causes (due to BW collider) for any stratum of BW
Example: for BW=’low, if we compare smoking vs. non-smoking we are also comparing rare other causes vs. likely other causes

Perspective 2

Mediation context of previous: we want to know the mortality rate of babies e.g. smokers vs. non if BW controlled for
However here we have confounders, whereas before, the fundamental assumption was that there weren’t any.
Ajusting for BW doesn’t sever all paths traversing the mediator, and actually opens up a new path, and the effect is now spurious
For low BW, comparison of smoking vs. non-smoking compares no other causes vs. other causes

Lord’s Paradox

Michael Clark

2016-05-18

Sex differences in weight gain (original Lord formulation)

Variables

Issues

Results

DAG

Data Example

Aside

Treatment with confounding (Wainer & Brown)

Variables

Issues

Results

Birth Weight Paradox

Variables

Issues

Results

Results

Collider bias (explain away effect)

Explanation

Perspective 1

Perspective 2